The data in the following table refers to a survey of
students in their final year of high school from a non-urban area near Dayton
Ohio USA. The survey asked the students whether they had ever used alcohol, 
cigarettes or marijuana. The given table also classifies the students by gender. 

\vspace{5mm}

\begin{center}
\begin{tabular}{ccccrcrcrcr}
 & & \multicolumn{2}{c}{Gender (G)}  & \multicolumn{3}{c}{Female} & & \multicolumn{3}{c}{Male}  
                               \\[2pt] \cline{5-7} \cline{9-11} 
\\[-5pt]
 &    & &  \multicolumn{7}{c}{\phantom{xxxxxxx} Marijuana Use (M)}  \\[2pt] \cline{5-11}
\\[-20pt]
 Alcohol   &    & Cigarette \\[-1pt]
Use (A) &  & Use (C) &\phantom{x}  & Yes & & No &\phantom{xx} & Yes & & No \\[2pt] \hline
\\[-5pt]
Yes &  &  Yes & & 428 & & 291 & & 483 & & 247  \\
    &  &  No  & & 15 & & 237 & & 29 & & 219     \\
No  &  &  Yes & & 1 & & 18 & & 2 & & 25        \\
    &  &  No  & & 1 & & 129 & & 1 & & 140      \\ \hline
\end{tabular}
\end{center}

\vspace{15mm}


\begin{enumerate}
\item{}
An initial analysis of the data produced the following association 
graph:

\vspace{5mm}

\setlength{\unitlength}{1mm}
\begin{picture}(110,50)(-30,0)
\put(10,40){\makebox(0,0){\LARGE A}}
\put(40,40){\makebox(0,0){\LARGE C}}
% \put(70,40){\makebox(0,0){\LARGE D}}
\put(40,10){\makebox(0,0){\LARGE M}}
\put(70,10){\makebox(0,0){\LARGE G}}
\thicklines
\put(15,40){\line(1,0){20}}
% \put(45,40){\line(1,0){20}}
\put(45,10){\line(1,0){20}}
\put(40,15){\line(0,1){20}}
\put(13.5,36.5){\line(1,-1){23}}
% \put(43.5,13.5){\line(1,1){23}}
\end{picture}


\vspace{5mm}


This graph was created by first fitting a Poisson regression model 
to the data and then writing down the association graph based on the 
identified model.

\vspace{5mm}

\begin{enumerate}
\item
The Poisson regression model for this data only contained main effects and 
2-way interactions. 
Given this information, what  model
must have been identified in order to produce the given association graph?
\item 
Give an example of a second Poisson regression model that could have produced 
this association graph (note you are not restricted to main effects and 
2-way interactions for this model).
\end{enumerate}

\marks{5}

\newpage

\item
What does the association graph indicate about the relationship 
between:
\begin{enumerate}
\item gender (G) and marijuana use (M) 
\item gender (G) and alcohol use (A).
\end{enumerate}
In each case state the nature of the relationship (independent, conditionally 
independent or directly related) and then explain what that means in terms of 
this data.

\marks{5}

\vspace{25mm}

\item
If we collapse this table on alcohol use (A) and cigarette use (C) we get 
the following 2-way table:
\begin{center}
\begin{tabular}{ccrcr}
& &  \multicolumn{3}{c}{\phantom{x} \underline{Marijuana Use}}  \\
Gender & \phantom{x} & Yes & & No \\[2pt] \hline
\\[-5pt]
Female   &  &  445 & & 675  \\
Male     &  &  515 & & 631  \\
\end{tabular}
\end{center}

Use this table to find a 95\% confidence interval for the {\em odds ratio} 
that compares the odds of marijuana use for females to the odds for males.
Note that if $Z\sim\mbox{N}(0,\,1)$, then $\mbox{Pr}(-1.96\le Z \le 1.96)=0.95$.


\marks{5}

\vspace{25mm}

\item 
The main objective of this study was to determine how marijuana use 
was related to alcohol use, to cigarette use, and to gender.
\begin{enumerate}
\item
Was it sensible to collapse the original table into the 2-way table given 
in part (b) to focus on the relationship between marijuana use and 
gender? Explain your response.
\item
Explain how you would investigate the relationship between marijuana use and 
alcohol use and the relationship between marijuana use and cigarette use.
Note that you are not required to actually investigate these relationships - just 
explain how you would go about it.
\end{enumerate}

\marks{5}


\end{enumerate}






